Mr. Ward's Teaching Materials

128i - Simulating Compound Events with Digital Tools

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Lesson 128 — Running Simulations with Digital Tools for Compound Events

Strand: Probability | Descriptor: AC9M8P03 | Duration: 45 minutes

Learning Intentions

  • To use digital tools, such as spreadsheet random-number functions, to simulate chance events efficiently at scale.
  • To set up simulation rules that correctly match a real event’s probabilities, including for compound events.

Success Criteria

I can:

  1. Translate a chance event into a digital simulation rule, assigning number ranges to outcomes correctly.
  2. Run a large number of simulated trials using a digital tool and record the results.
  3. Calculate relative frequency from simulated data, including for compound (“and”) conditions.
  4. Explain why digital simulation allows many more trials than a hands-on experiment, and why this matters.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

  1. What values can the spreadsheet formula =RANDBETWEEN(1,6) produce?
  2. Are all six values in Q1 equally likely?
  3. How could =RANDBETWEEN(1,2) be used to simulate a fair coin toss?
  4. What does the function COUNTIF do, in your own words?

Answers: 1. Any whole number from to inclusive; 2. Yes, each is equally likely; 3. Let heads, tails; 4. It counts how many cells in a range meet a stated condition.

Activities

Activity 1 — Explicit Instruction: Building a Simulation Rule (12 min)

I do / We do / You do.

I do: Simulate a fair coin. Formula: =RANDBETWEEN(1,2), with heads, tails. To count heads across simulated tosses in column A: =COUNTIF(A1:A200,1). Relative frequency of heads that count .

We do: Together design a rule for a spinner with unequal sectors out of : red covers –, blue covers –, green covers –. Formula: =RANDBETWEEN(1,10). Check the proportions match the sectors: red , blue , green .

You do: Design the number ranges for =RANDBETWEEN(1,20) to simulate a spinner with outcomes A (), B (), C ().

(Answer: A – (), B – (), C – ().)

Activity 2 — Applied Task: Simulating Two Dice (16 min)

Pairs, working with a spreadsheet (or a provided printed simulation output if a device is unavailable).

Recall from Lesson 122 that rolling two fair dice gives and .

Set-up:

ColumnFormula
A (Die 1)=RANDBETWEEN(1,6)
B (Die 2)=RANDBETWEEN(1,6)
C (Sum)=A1+B1
D (Same?)=IF(A1=B1,1,0)

Fill down rows, then:

Sample simulated output (for discussion): sum occurred times; both the same occurred times.

Compare with the theoretical for each — both simulated values sit reasonably close.

Now extend — a genuinely compound (“and”) condition: find using =COUNTIFS(A1:A300,6,B1:B300,6)/300. State the theoretical value to check against.

(Theoretical: . COUNTIFS is required here, not two separate COUNTIFs, since the condition needs both columns to be true on the same row.)

Activity 3 — Inquiry: why Scale Matters (7 min)

Whole class discussion.

A hands-on experiment of dice rolls takes real time and effort. A spreadsheet can simulate rolls in under a second.

  1. Why might a result from simulated rolls still swing quite far from the theoretical probability?
  2. What would you expect to happen to that swing at rolls? At ?
  3. Why is a digital tool especially useful for exploring this question?

Discussion points: Small samples are more strongly affected by chance swings; a digital tool lets you generate thousands of trials almost instantly, letting the class see the swings shrink as the sample grows, rather than just being told they should. (This becomes the formal focus of Lesson 129 — the law of large numbers.)

Checks for Understanding

(5 minutes — exit ticket, collected)

  1. Write the spreadsheet formula to simulate rolling a fair -sided die.
  2. A spinner has outcomes A (), B (), C (). Using =RANDBETWEEN(1,20), state the number ranges you would assign to each outcome.
  3. After simulating trials of two coins tossed together, COUNTIF found “both heads” times. Find the simulated relative frequency, and compare it with the theoretical value of .
  4. Give one advantage of digital simulation over a hands-on physical experiment for estimating a probability.

Answers: 1. =RANDBETWEEN(1,8); 2. A –, B –, C –; 3. , slightly below the theoretical ; 4. It allows far more trials in far less time, giving a more stable and reliable estimate.

Common Misconceptions

MisconceptionHow to pre-empt it
Assigning unequal-sized number ranges to outcomes that should be equally likely.Always check that each outcome’s range contains the same count of integers, e.g. – and – for two equally likely outcomes from RANDBETWEEN(1,6).
Using RANDBETWEEN(1,6) but believing it can also return or .State explicitly that both endpoints are included and no other values are possible.
Using two separate COUNTIFs and adding them for an “and” condition.Model COUNTIFS directly, contrasting it with the “or” case, which does use two COUNTIFs added together.
Believing a simulated relative frequency must exactly match the theoretical probability, however many trials are run.Revisit Activity 2’s sample output — close, but not exact, even at trials.
Forgetting to divide the COUNTIF/COUNTIFS result by the total number of simulated trials.Insist relative frequency is always written as a fraction of the total, never as a raw count.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A simulation uses =RANDBETWEEN(1,100) to model an event with probability . Which range of integers should represent “the event occurs”?

Answer

Any consecutive integers within –, e.g. –.

E2 (Kangaroo style). Two dice are simulated times. COUNTIF finds that “sum ” occurred times. Find the simulated relative frequency and compare it with the theoretical value of .

Answer

Close — a difference of about .

E3 (Challenge). Using the two-dice simulation columns from Activity 2, describe the formulas needed to estimate , and state the theoretical value (from Lesson 121) to check against.

Answer

Add a column, e.g. =IF(OR(A1=6,B1=6),1,0), then find =SUM(that column)/300. Theoretical value: .

Homework

  1. Write the spreadsheet formula to simulate a fair coin toss, stating which value represents heads.
  2. A spinner has outcomes X (), Y (), Z (). Using =RANDBETWEEN(1,20), assign a number range to each outcome.
  3. COUNTIF is used on simulated die rolls and finds face occurred times. Find the simulated relative frequency and compare it with the theoretical .
  4. Explain the difference between COUNTIF and COUNTIFS, using a probability example for each.
  5. Reasoning. A student sets up =RANDBETWEEN(0,5) to simulate a fair die. Explain what is wrong with this formula.
  6. Reasoning. Explain why running simulated trials gives a more reliable relative frequency than running .
  7. Challenge. Design the RANDBETWEEN ranges needed to simulate a spinner with outcomes in the ratio (total parts), and state the probability each range represents.

Answers: 1. =RANDBETWEEN(1,2), with heads (or equivalent); 2. X –, Y –, Z –; 3. , compared with — close, slightly higher; 4. COUNTIF counts cells meeting one condition (e.g. how many rolls were a ); COUNTIFS counts rows meeting two or more conditions simultaneously (e.g. how many rows had Die 1 and Die 2 ); 5. RANDBETWEEN(0,5) produces possible values (), which happens to still be equally likely outcomes and could be relabelled to work — but it does not match the natural labels – on a real die, risking confusion when interpreting results; 6. Larger samples average out chance swings more effectively, so the relative frequency settles closer to the true probability — this is explored fully in Lesson 129; 7. Using =RANDBETWEEN(1,10): outcome A – (), B – (), C – (), D (), E ().

Graph View

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